7.1: Recursive and Explicit Formulas

Learning objectives

Write a recursive formula for a sequence, and use the formula to identify terms in the sequence.

Write an explicit formula for a sequence, and use the formula to identify terms in the sequence.

Identify a sequence as arithmetic, geometric, or neither.

Introduction

Consider a situation in which the value of a car depreciates 10% per year. If the car is originally valued at $20,000, the following year it is worth 90% of $20,000, or $18,000. After another, the value is 90% of $18,000, or $16,200. If we write the decreasing values as a list: 20,000, 18,000, 16,200 ... we have written a sequence.

A sequence is an ordered list of objects. Above our "objects" were all numbers. The simplest way to represent a sequence is by listing some of its terms. For example the sequence of odd, positive integers is shown here:

1, 3, 5, 7 ...

In this lesson you will learn to represent a sequence in two ways. The first method requires that you know the previous term in order to find the next term in the sequence. The second method does not.

Representing a Sequence Recursively

Consider again the sequence shown above. What is the next term?

As long as you are familiar with the odd integers (i.e., you can count in 2’s) you can figure out that the next term is 9. If we want to describe this sequence in general, we can do so by stating what the first term is, and then by stating the relationship between successive terms. When we represent a sequence by describing the relationship between its successive terms, we are representing the sequence recursively.

The terms in a sequence are often denoted with a variable and a subscript. All of the terms in a given sequence are written with the same variable, and increasing subscripts. So we might list terms in a sequence as a1, a2, a3, a4, a5 ...

We can use this notation to represent the example above. This sequence is defined as follows:

a1 = 1

an = an-1 + 2

At first glance this notation may seem confusing. What is important to keep in mind is that the subscript of a term represents its “place in line.” So an just means the nth term in the sequence. The term an-1 just means the term before an. In the sequence of odd numbers above, a1 = 1, a2 = 3, a3 = 5, a4 = 7, a5 = 9 and so on. If, for example, we wanted to find a10, we would need to find the 9th term in the sequence first. To find the 9th term we need to find the 8th term, and so on, back to a term that we know.

Finding terms in this sequence is relatively straightforward, as the pattern is familiar. However, this would clearly be tedious if you needed to find the 100th term. We will turn to another method of defining sequences shortly. First let’s consider some more complicated sequences.

Example 2: For each sequence, find the indicated term.

a. Find the 5th term for the sequence:

t1 = 3

tn = 2tn-1

b. Find the 4th term for the sequence:

b1 = 3

bn = (bn-1)2 + 1

Solution:

a. t5 = 48

t2 = 2t1 = 2 × 3 = 6

t1 = 3

t3 = 2t2 = 2 × 6 = 12

tn = 2 × tn-1

t4 = 2t3 = 2 × 12 = 24

t5 = 2t4 = 2 × 24 = 48

b. b4 = 677

b2 = (b1)2 + 1 = 22 + 1 = 4 + 1 = 5

b1 = 2

b3 = (b2)2 + 1 = 52 + 1 = 25 + 1 = 26

bn = (bn-1)2 + 1

b4 = (b3)2 + 1 = 262 + 1 = 676 + 1 = 677

As you can see from just a few terms of the sequence in example 2b, the terms in a sequence can grow quickly. If we compare the growth of the terms in the sequences we have seen so far, the first example, the sequence of odd numbers, was the slowest. Its growth is linear, and it is referred to as an arithmetic sequence. Every arithmetic sequence has a common difference, or a constant difference between each term. (The common difference is analogous to the slope of a line.) The sequence of odd numbers has a common difference of 2 because for all n, an - an- 1 = 2. The sequence in example 2a is a geometric sequence. Every geometric sequence has a common ratio. In the sequence in example 2a, the common ratio is 2 because for all n, . The terms of a geometric sequence follow an exponential pattern. The sequence in example 2b is neither arithmetic nor geometric, though its values follow a cubic pattern.

For any of these sequences, as noted above, determining more than a few values by hand would be time consuming. Next you will learn to define a sequence in a way that makes finding the nth term faster.

Representing a sequence explicitly

When we represent a sequence with a formula that lets us find any term in the sequence without knowing any other terms, we are representing the sequence explicitly.

Given a recursive definition of an arithmetic or geometric sequence, you can always find an explicit formula, or an equation to represent the nth term of the sequence. Consider for example the sequence of odd numbers we started with: 1,3,5,7,...

We can find an explicit formula for the nth term of the sequence if we analyze a few terms:

a1 = 1

a2 = a1 + 2 = 1 + 2 = 3

a3 = a2 + 2 = 1 + 2 + 2 = 5

a4 = a3 + 2 = 1 + 2 + 2 + 2 = 7

a5 = a4 + 2 = 1 + 2 + 2 + 2 + 2 = 9

a6 = a6 + 2 = 1 + 2 + 2 + 2 + 2 + 2 = 11

Note that every term is made up of a 1, and a set of 2’s. How many 2’s are in each term?

a1

= 1

a2

= 1 + 2 = 3

a3

= 1 + 2 × 2 = 5

a4

= 1 + 3 × 2 = 7

a5

= 1 + 4 × 2 = 9

a6

= 1 + 5 × 2 = 11

The nth term has (n - 1)2 ’s. For example, a99 = 1 + 98 × 2 = 197 . We can therefore represent the sequence as an = 1 + 2(n - 1). We can simplify this expression:

an

=1+2(n-1)

an

=1 + 2n - 2

an

=2n-1

In general, we can represent an arithmetic sequence in this way, as long as we know the first term and the common difference, d. Notice that in the previous example, the first term was 1, and the common difference, d, was 2. The nth term is therefore the first term, plus d(n-1):

an

=a1+d(n-1)

You can use this general equation to find an explicit formula for any term in an arithmetic sequence.

Example 3: Find an explicit formula for the nth term of the sequence 3,7,11,15... and use the equation to find the 50th term in the sequence.

Solution:an=4n-1 , and a50=199

The first term of the sequence is 3, and the common difference is 4.

an

=a1+d(n-1)

an

=3+4(n-1)

an

=3+4n-4

an

=4n-1

a50

=4(50)-1=200-1=199

We can also find an explicit formula for a geometric sequence. Consider again the sequence in example 2a:

t2 = 2t1 = 2 × 3 = 6

t1 = 3

t3 = 2t2 = 2 × 6 = 12

tn = 2 × tn-1

t4 = 2t3 = 2 × 12 = 24

t5 = 2t4 = 2 × 24 = 48

Notice that every term is the first term, multiplied by a power of 2. This is because 2 is the common ratio for the sequence.

t1

=3

t2

=2 × 3 = 6

t3

=2 × 2 × 6 = 22 × 6 = 12

t4

=2 × 2 × 2 × 6=23 × 6 = 24

t5

=2 × 2 × 2 × 26 = 24 × 6 = 48

The power of 2 in the nth term is (n-1). Therefore the nth term in this sequence can be defined as: tn=3(2n-1). In general, we can define the nth term of a geometric sequence in terms of its first term and its common ratio, r:

tn

=t1(rn-1)

You can use this general equation to find an explicit formula for any term in a geometric sequence.

Example 4: Find an explicit formula for the nth term of the sequence 5,15,45,135... and use the equation to find the 10th term in the sequence.

Solution:an=5 × 31,and a10 = 98,415

The first term in the sequence is 5, and r = 3.

an

=a1 × rn-1

an

=5 × 3n-1

a10

=5 × 310-1

a10

=5 × 39 = 5 × 19,683 = 98,415

Again, it is always possible to write an explicit formula for terms of an arithmetic or geometric sequence. However, you can also write an explicit formula for other sequences, as long as you can identify a pattern. To do this, you must remember that a sequence is a function, which means there is a relationship between the input and the output. That is, you must identify a pattern between the term and its index, or the term’s “place” in the sequence.

Example 5: Write an explicit formula for the nth term of the sequence 1,(1/2),(1/3),(1/4)...

Solution:an=(1/n)

Initially you may see a pattern in the fractions, but you may also wonder about the first term. If you write 1 as (1/1), then it should become clear that the nth term is (1/n).

Lesson Summary

In this lesson you have learned to represent a sequence in three ways. First, you can represent a sequence just by listing its terms. Second, you can represent a sequence with a recursive formula. Given a list of terms, writing a formula requires stating the first term and the relationship between successive terms. Finally, you can write a sequence using an explicit formula. Doing this requires identifying a pattern between n and the nth term.

In this lesson we have looked at several kinds of sequences: arithmetic, geometric, and sequences that do not follow either pattern. In the remainder of the chapter you will again see these three general categories of sequences.

Points to Consider

What is the difference between a recursive and an explicit representation of a sequence?

How many terms do sequences have?

What happens if we add up the terms in a sequence?

Review Questions

Find the value of a6 , given the sequence defined as:

a1

=4

an

=5an-1

Find the value of a5, given the sequence defined as:

a1

=32

an

=(1/2)an-1

Find the value of an-1 , given the sequence defined as:

a1

=1

an

=3an-1-n

Consider the sequence: 2,9,16... Write an explicit formula for the sequence, and use the formula to find the value of the 20th term.

Consider the sequence: 5,10,20... Write an explicit formula for the sequence, and use the formula to find the value of the 9th term.

Consider the sequence (1/2)(1/4)(1/8) Write an explicit formula for the sequence, and use the formula to find the value of the 7th term.

Identify all sequences in the previous six problems that are geometric. What is the common ratio in each sequence?

Consider the situation in the introduction: a car that is originally valued at $20,000 depreciates by 10% per year. What kind of sequence is this? What is the value of the car after 10 years?

In a particular arithmetic sequence, the second term is 4 and the fifth term is 13. Write an explicit formula for this sequence.

The membership of an online dating service increases at an average rate of 8% per year. In the first year, there are 500 members. a. How many members are there in the second year? b. How many members are there in the eighteenth year?

Review Answers

The 6th term is 12,500

The 5th term is 2

The 7th term is -178

an

=7n-5

a20

=135

an

=5 × 2n-1

a9

=1280

The sequence in question 1 has r = 5. The sequence is question 2 has r= 1/2. The sequence in question 5 has r = 2. The sequence in question 6 has r= 1/2.

The sequence is a geometric sequence. The value of the car after 10 years is approximately $7748.

an=3n-2

a. 540 members b. Approximately 1,998 members

Vocabulary

Explicit formula

An explicit formula for a sequence allows you to find the value of any term in the sequence.

Natural numbers

The natural numbers are a subset of the integers: {1,2,3,4,5....}

Recursive formula

A recursive formula for a sequence allows you to find the value of the nth term in the sequence if you know the value of the(n-1)th term in the sequence.